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Large order Reynolds expansions for the Navier-Stokes equations

We consider the Cauchy problem for the incompressible homogeneous Navier-Stokes (NS) equations on a d-dimensional torus, in the C^infinity formulation described, e.g., in [25]. In [22][25] it was shown how to obtain quantitative estimates on the exact solution of the NS Cauchy problem via the "a posteriori" analysis of an approximate solution; such estimates concern the interval of existence of the exact solution and its distance from the approximate solution. In the present paper we consider an approximate solutions of the NS Cauchy problem having the form u^N(t) = sum_{j=0}^N R^j u_j(t), where R is the "mathematical" Reynolds number (the reciprocal of the kinematic viscosity) and the coefficients u_j(t) are determined stipulating that the NS equations be satisfied up to an error O(R^{N+1}). This subject was already treated in [24], where, as an application, the Reynolds expansion of order N=5 in dimension d=3 was considered for the initial datum of Behr-Necas-Wu (BNW). In the present paper, these results are enriched regarding both the theoretical analysis and the applications. Concerning the theoretical aspect, we refine the approach of [24] following [25] and use the symmetries of the initial datum in building up the expansion. Concerning the applicative aspect we consider two more (d=3) initial data, namely, the vortices of Taylor-Green (TG) and Kida-Murakami (KM); the Reynolds expansions for the BNW, TG and KM data are performed via a Python program, attaining orders between N=12 and N=20. Our a posteriori analysis proves, amongst else, that the solution of the NS equations with anyone of the above three data is global if R is below an explicitly computed critical value. Our critical Reynolds numbers are below the ones characterizing the turbulent regime; however these bounds have a sound theoretical support, are fully quantitative and improve previous results of global existence.